They were the descendants of Phoenician settlers. Phoenicians were a Semitic civilization in today's Lebanon. They were excellent seafarers and traders, founding settlements in many places along the coast of the Mediterranean Sea towards the Iberian Peninsula and Northwest Africa. When you read about this era, the Punic peoples are a cultural-linguistic identity that is descendant from the Phoenicians.

If you want something a little more comprehensive about Carthage and the Punic people (instead of just the Punic Wars) - there's a lot more focus on Carthage in the book Carthage Must Be Destroyed by Richard Miles. I'm not a historian though, so maybe someone has a better suggestion. read what /u/ScipioAsina says as he actually has the flair